Irreducibility of a general polynomial in a finite field

[Mystic998]

Homework Statement

For prime p, nonzero $a \in \bold{F}_p$, prove that $q(x) = x^p - x + a$ is irreducible over $\bold{F}_p$.

Homework Equations

The Attempt at a Solution

It's pretty clear that none of the elements of $\bold{F}_p$ are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if $\alpha$ is a root of q(x), then so is $\alpha + 1$, and from there I was able to deduce (hopefully not incorrectly) that $\bold{F}_{p}(\alpha)$ is the splitting field for q(x) over the field, but I can't figure out the degree of the extension, so I'm kind of stuck. Any ideas?

 

[morphism]

How about showing that if q(x) is reducible, then it must split into linear factors?

 

[Mystic998]

I sat and thought about it for about 45 minutes and played around on a chalkboard with it, and I'm not sure how to go about your suggestion. Is it related to the work I already did, or is it a completely new line of attack? Thanks for helping.

 

[Mystic998]

Okay, I figured it out. Thanks a lot.